Download Common Factoring

polynomials
polynomials
Distributive Law
MPM2D: Principles of Mathematics
Recap
Expand and simplify 2x(x 2 − 4x + 5).
Factoring Polynomials
2x(x 2 − 4x + 5) = 2x · x 2 − 2x · 4x + 2x · 5
Part 1: Common Factoring
= 2x 3 − 8x 2 + 10x
J. Garvin
J. Garvin — Factoring Polynomials
Slide 2/11
Slide 1/11
polynomials
polynomials
Common Factoring
Common Factoring
Recall that the Greatest Common Factor (GCF) of two or
more numbers is the greatest value that divides evenly into
those numbers.
Consider the polynomial expression 6x + 15.
For example, the GCF of 6 and 9 is 3, since both 6 and 9 are
divisible by 3, but not by a greater number.
Similarly, the GCF of x 3 and x 5 is x 3 , since x 3 is divisible by
x5
x 3 (obviously), as is x 5 : 3 = x 2 .
x
The process of rewriting a polynomial expression as a
product of two or more factors is called factoring.
To rewrite the expression in factored form, express it as a
product of 3 and some other polynomial. This polynomial is
determined by dividing each term by the GCF.
6x
15
6x + 15 = 3
+
3
3
= 3(2x + 5)
If a factor is common to all terms, then we can common
factor the expression.
Note that the common factor is never discarded. It remains
as part of the factored expression.
The GCF of both 6x and 15 is 3, since both 6x and 15 are
evenly divisible by 3, but not by a greater number.
The Distributive Law can be used to validate the solution.
J. Garvin — Factoring Polynomials
Slide 3/11
J. Garvin — Factoring Polynomials
Slide 4/11
polynomials
polynomials
Common Factoring
Common Factoring
Example
A common factor must apply to all terms in a polynomial
expression.
Factor 8x − 12.
The GCF of 8x and −12 is 4, so the factored expression will
be the product of 4 and another polynomial.
8x
12
8x − 12 = 4
−
4
4
= 4(2x − 3)
Again, note that the common factor, 4, remains as part of
the factored expression.
J. Garvin — Factoring Polynomials
Slide 5/11
Example
Factor 3x 2 − 15x + 21.
The GCF of 3x 2 , −15x and 21 is 3, so the factored
expression will be the product of 3 and another polynomial.
2
3x
15x
21
3x 2 − 15x + 21 = 3
−
+
3
3
3
= 3(x 2 − 5x + 7)
J. Garvin — Factoring Polynomials
Slide 6/11
polynomials
polynomials
Common Factoring
Common Factoring
Sometimes it is possible to factor a variable from each term
of a polynomial.
In other cases, both a variable and a constant make up the
GCF.
Example
Example
Factor
5x 3
−
2x 2
Factor 15x 4 + 25x 3 − 30x.
+ 3x.
The GCF of the three terms is x, so the factored expression
will be the product of x and another polynomial.
3
5x
2x 2 3x
5x 3 − 2x 2 + 3x = x
−
+
x
x
x
The GCF of the three terms is 5x. Note that only a single x
can be factored, since the exponent on the last term is 1.
15x 4 25x 3 30x
+
−
15x 4 + 25x 3 − 30x = 5x
5x
5x
5x
J. Garvin — Factoring Polynomials
Slide 7/11
J. Garvin — Factoring Polynomials
Slide 8/11
= x(5x 2 − 2x + 3)
= 5x(3x 3 + 5x 2 − 6)
polynomials
polynomials
Common Factoring
Common Factoring
Expressions involving multiple variables can be handled in the
same way.
Example
Example
The GCF of the three terms is 1. Therefore, the expression
cannot be common factored.
Factor 16x 3 y 2 + 24x 5 y 2 .
While it is not always possible to find a common factor for
the three terms, there are other ways in which polynomials
like the one above can be factored.
The GCF of the two terms is 8x 3 y 2 . Again, the factor is
limited by the lowest exponent on a given variable.
16x 3 y 2 24x 5 y 2
16x 3 y 2 + 24x 5 y 2 = 8x 3 y 2
+
3
2
3
2
8x y
8x y
For example, note that the expression can be factored as
(2x − 1)(x + 3) = 2x 2 + 6x − x − 3 = 2x 2 + 5x − 3.
= 8x 3 y 2 (2 + 3x 2 )
Note that there are no terms containing y inside of the
brackets, since y 2 was completely factored out of all terms.
J. Garvin — Factoring Polynomials
Slide 9/11
J. Garvin — Factoring Polynomials
Slide 11/11
We will explore some of these other methods over the next
few lessons.
J. Garvin — Factoring Polynomials
Slide 10/11
polynomials
Questions?
Factor 2x 2 + 5x − 3.